Finite Group Databasebeta

← #36336153

Subgroups — #36336153

Order 21376.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D668C211yesmaximal, characteristic
2#24923635C211yesmaximal, characteristic, complemented
3#24923593C211yesmaximal, characteristic, complemented
4#24923582C211yesmaximal, characteristic, complemented
5#24923566C211yesmaximal, characteristic, complemented
6#24923517C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923469C211yesmaximal, characteristic, complemented
9#24923419C211yesmaximal, characteristic, complemented
10#24923419C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923273C211yesmaximal, characteristic, complemented
14C2×#19128752C211yesmaximal, characteristic, complemented
15#552×C167C211yesmaximal, characteristic, complemented
16C22×D668C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×D4×D167—21no—
19C2×#14581937C2211yescharacteristic
20C2×#14581937C2211yescharacteristic, complemented
21C2×#14581937C2211yescharacteristic, complemented
22C2×#14581934C2211yescharacteristic, complemented
23C22×C167⋊C8C2211yescharacteristic, complemented
24#19128829C2211yescharacteristic
25#19128807—21no—
26#19128806C2211yescharacteristic
27#19128806C2211yescharacteristic
28#19128802—21no—
29#19128802—21no—
30C4×D668C2211yescharacteristic
31#19128765C2211yescharacteristic, complemented
32#19128765C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128756C2211yescharacteristic, complemented
36#19128752C2211yescharacteristic
37#19128752C2211yescharacteristic
38#19128752C2211yescharacteristic
39#19128752C2211yescharacteristic, complemented
40#19128752C2211yescharacteristic, complemented
41#19128752C2211yescharacteristic, complemented
42#19128752C2211yescharacteristic, complemented
43#19128752C2211yescharacteristic, complemented
44#19128752C2211yescharacteristic, complemented
45C22⋊(C167⋊C8)C2211yescharacteristic, complemented
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48#126×C167—21no—
49#125×C167—21no—
50#125×C167C2211yescharacteristic, complemented
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic, complemented
53C22⋊Q8×C167C2211yescharacteristic, complemented
54C4×Q8×C167C2211yescharacteristic, complemented
55C4×D4×C167C2211yescharacteristic
56#117×C167—21no—
57#117×C167C2211yescharacteristic, complemented
58C2×C4⋊C4×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C23×D167—21no—
61C2×D668—22no—
62C2×D668—22no—
63C2×D668—21no—
64C2×D668—21no—
65C2×D668—21no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668—41no—
69C2×D668C2311yescharacteristic
70C2×D668C2311yescharacteristic
71C2×D668C2311yescharacteristic
72C2×C22⋊D167—21no—
73C2×C22⋊D167—21no—
74D4×D167—44no—
75C2×C4×D167—21no—
76#14581937—22no—
77#14581937—41no—
78#14581937—41no—
79#14581934—41no—
80C2×C167⋊C8—41no—
81C2×C167⋊C8C2311yescharacteristic
82C2×C167⋊C8C2311yescharacteristic
83C2×C167⋊C8C2311yescharacteristic
84C2×C167⋊C8C2311yescharacteristic
85C4⋊Dic167C2311yescharacteristic
86(C2×D167)⋊C4—21no—
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167D411yescharacteristic
91C22×C4×C167—21no—
92C22×C4×C167C2311yescharacteristic
93C22×C4×C167C2311yescharacteristic
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167—21no—
98C4⋊C4×C167—21no—
99C4⋊C4×C167—21no—
100C4⋊C4×C167D411yescharacteristic
101C4⋊C4×C167C2311yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
109C42×C167—21no—
110C42×C167—21no—
111C42×C167C2311yescharacteristic
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—21no—
115C22×D167—42no—
116C22×D167—42no—
117C22×D167—42no—
118C22×D167—42no—
119C22×D167—81no—
120C22×D167—81no—
121D668—21no—
122D668—21no—
123D668—21no—
124D668—21no—
125D668—21no—
126D668—42no—
127D668—42no—
128D668—42no—
129D668—81no—
130D668#4112yes—
131C22⋊D167—42no—
132C22⋊D167—42no—
133C2×Dic167—21no—
134C4×D167—42no—
135C167⋊C8—21no—
136C167⋊C8—21no—
137C167⋊C8—21no—
138C167⋊C8—21no—
139C23×C167C2×D411yescharacteristic
140C23×C167C2×D411yescharacteristic
141Q8×C167—21no—
142Q8×C167—21no—
143D4×C167—22no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—42no—
153C2×C4×C167C2×D411yescharacteristic
154C2×C4×C167C2×D411yescharacteristic
155C2×C4×C167C2×D411yescharacteristic
156C2×C4×C167C2×D411yescharacteristic
157C2×C4×C167C2411yescharacteristic
158C22×C167—21no—
159C22×C167—21no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—21no—
163C22×C167AGL(1,ℤ/8)12yes—
164C22×C167C22×D411yescharacteristic
165C2×D167—22no—
166C2×D167—41no—
167C2×D167—41no—
168C2×D167—41no—
169C2×D167—41no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—84no—
175C2×D167—84no—
176C4×C167—21no—
177C4×C167—21no—
178C4×C167—21no—
179C4×C167—21no—
180C4×C167—41no—
181C4×C167—41no—
182C4×C167—41no—
183C4×C167—41no—
184C4×C167C2×#4111yescharacteristic
185C4×C167H(5,𝔽2)11yescharacteristic
186Dic167—41no—
187C2×C167—22no—
188C2×C167—41no—
189C2×C167#54611yescharacteristic
190C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C167#57711yescharacteristic
192D167—42no—
193D167—81no—
194D167—81no—
195C167#331011yescharacteristic, complemented, Sylow
196#3310—1671nomaximal, Sylow
197#552D16711yescharacteristic, complemented
198D42—1671no—
199#490—1671no—
200#482—1671no—
201#480—1671no—
202#478—1671no—
203#468—1671no—
204#465—1671no—
205#460—1671no—
206#450—1671no—
207#450—1671no—
208#440—1671no—
209#435—1671no—
210#418—1671no—
211C2×#102—1671no—
212C22×D4—3341no—
213C22×D4—1671no—
214C22×D4—1671no—
215C2×SD16—1671no—
216C2×SD16—1671no—
217C2×SD16—1671no—
218C2×M16—1671no—
219C22×C8—1671no—
220#127—1671no—
221#126—21no—
222#125—21no—
223#125C2×D16711yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D16711yescharacteristic, complemented
226C22⋊Q8C2×D16711yescharacteristic, complemented
227#121—3341no—
228#121—1671no—
229#121—1671no—
230#120—3341no—
231#120—3341no—
232C4×Q8C2×D16711yescharacteristic, complemented
233C4×D4—1671no—
234C4×D4C2×D16711yescharacteristic
235#117—21no—
236#117C2×D16711yescharacteristic, complemented
237C2×C4⋊C4C2×D16711yescharacteristic, complemented
238#107—1671no—
239#107—1671no—
240#106—1671no—
241C4⋊C8—1671no—
242#103—1671no—
243#102—1671no—
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102—1671no—
251#102—1671no—
252C22⋊C8—1671no—
253C22⋊C8—1671no—
254#97—1671no—
255C24—3341no—
256C24—3341no—
257C2×Q8C22×D16711yescharacteristic
258C2×D4—3342no—
259C2×D4—3342no—
260C2×D4—3341no—
261C2×D4—3341no—
262C2×D4—3341no—
263C2×D4—3341no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—6681no—
268C2×D4—6684no—
269C2×D4—1671no—
270C2×D4—1671no—
271C2×D4—1671no—
272C2×D4C22⋊D16711yescharacteristic
273C22×C4—21no—
274C22×C4—3341no—
275C22×C4C22×D16711yescharacteristic
276C22×C4C22×D16711yescharacteristic
277SD16—3342no—
278SD16—6681no—
279SD16—6681no—
280M16—6681no—
281C2×C8—6681no—
282C2×C8—1671no—
283C2×C8—1671no—
284C2×C8—1671no—
285C2×C8—1671no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1671no—
293C4⋊C4C22⋊D16711yescharacteristic
294C4⋊C4C22×D16711yescharacteristic
295C4⋊C4C22×D16711yescharacteristic
296C4⋊C4C22×D16711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4C22⋊D16711yescharacteristic
304C22⋊C4C22⋊D16711yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D16711yescharacteristic
308C23—13361no—
309C23—13361no—
310C23—3341no—
311C23—3341no—
312C23—3341no—
313C23—6682no—
314C23—6682no—
315C23—6682no—
316C23—6682no—
317C23C2×C22⋊D16711yescharacteristic
318C23C2×C22⋊D16711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—13361no—
323D4—3341no—
324D4—3341no—
325D4—3341no—
326D4—3341no—
327D4—3341no—
328D4—6682no—
329D4—6682no—
330D4—6682no—
331D4—6682no—
332D4—6682no—
333D4—1672no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3341no—
344C2×C4—6682no—
345C2×C4C2×C22⋊D16711yescharacteristic
346C2×C4C2×C22⋊D16711yescharacteristic
347C2×C4C2×C22⋊D16711yescharacteristic
348C2×C4C2×C22⋊D16711yescharacteristic
349C2×C4C23×D16711yescharacteristic
350C8—3341no—
351C8—3341no—
352C8—3341no—
353C8—3341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—13364no—
360C22—13364no—
361C22—3342no—
362C22—6681no—
363C22—6681no—
364C22—6681no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22#1912885712yes—
371C22C22×C22⋊D16711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—6681no—
381C4#41×D16711yescharacteristic
382C4#1912887811yescharacteristic
383C2—22no—
384C2—41no—
385C2—13361no—
386C2—13361no—
387C2—6682no—
388C2#2492390711yescharacteristic
389C2C2×#1912885711yescharacteristic
390C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#552×C167C2
socleC22×C167C22×D4
prime radicalD4×D668C2
hypercenter#552D167
hypocenterC167#3310
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #24923635C2semidirect
3. #24923593C2semidirect
4. #24923582C2semidirect
5. #24923566C2semidirect
6. #24923517C2semidirect
7. #24923500C2semidirect
8. #24923469C2semidirect
9. #24923419C2semidirect
10. #24923419C2semidirect
11. #24923367C2semidirect
12. #24923341C2semidirect
13. #24923273C2semidirect
14. C2×#19128752C2semidirect
15. #552×C167C2semidirect
20. C2×#14581937C22semidirect
21. C2×#14581937C22semidirect
22. C2×#14581934C22semidirect
23. C22×C167⋊C8C22semidirect
31. #19128765C22semidirect
32. #19128765C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128756C22semidirect
39. #19128752C22semidirect
40. #19128752C22semidirect
41. #19128752C22semidirect
42. #19128752C22semidirect
43. #19128752C22semidirect
44. #19128752C22semidirect
45. C22⋊(C167⋊C8)C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
50. #125×C167C22semidirect
52. C22⋊Q8×C167C22semidirect
53. C22⋊Q8×C167C22semidirect
54. C4×Q8×C167C22semidirect
57. #117×C167C22semidirect
58. C2×C4⋊C4×C167C22semidirect
195. C167#3310semidirect
197. #552D167semidirect
223. #125C2×D167semidirect
225. C22⋊Q8C2×D167semidirect
226. C22⋊Q8C2×D167semidirect
232. C4×Q8C2×D167semidirect
236. #117C2×D167semidirect
237. C2×C4⋊C4C2×D167semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D4×D6682
#249236352
#249235932
#249235822
#249235662
#249235172
#249235002
#249234692
#249234192
#249233672
#249233412
#249232732
C2×#191287522
#552×C1672
#3310167